Finite Group Databasebeta

← C2×#11722720 #13166408

Subgroups — C2×#11722720 #13166408

Order 1728.

376 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C2×C3⋊#3708C211yesmaximal, characteristic, complemented
3C2×C3×#3708C211yesmaximal, characteristic, complemented
4#11722720C214yesmaximal, direct product factor
5C2×#63773—31nomaximal
6C2×C32×C4×D3C411yescharacteristic
7C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9D3×C3⋊(C3⋊C8)C412yes—
10D3×C3⋊(C3⋊C8)C2212yes—
11C3⋊#3708C2212yescomplemented
12C3×#3708C2212yescomplemented
13C22×C3⋊(C3⋊C8)—31no—
14C2×D3×C3⋊C8—62no—
15#63773—34no—
16C2×#3708—31no—
17C2×#3708D311yescharacteristic, complemented
18C22×C32×D3C811yescharacteristic
19C2×C33×C4C2×C411yescharacteristic
20C2×C32×Dic3C811yescharacteristic
21C32×C4×D3C812yes—
22C32×C4×D3C2×C412yes—
23C3⋊(C3⋊(C3⋊C8))C2×C412yes—
24C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
25C3×C3⋊(C3⋊C8)C2311yescharacteristic
26C2×#798—91nomaximal
27C22×C32×C4—31no—
28C2×C3×C4×D3—22no—
29C2×C3⋊(C3⋊C8)—31no—
30C2×C3⋊(C3⋊C8)—32no—
31C2×C3⋊(C3⋊C8)—32no—
32C2×C3⋊(C3⋊C8)—62no—
33C2×C3⋊(C3⋊C8)—122no—
34C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
35C2×C3×C3⋊C8—62no—
36D3×C3⋊C8—64no—
37D3×C3⋊C8—64no—
38#3708—32no—
39#3708C2×D312yescomplemented
40C22×C33C2×C811yescharacteristic
41C2×C32×D3—22no—
42C2×C32×D3C2×C812yes—
43C33×C4C2×C811yescharacteristic
44C33×C4C22×C411yescharacteristic
45C32×Dic3C2×C812yes—
46C22×C3⋊C8—182no—
47C2×D3×C8—91no—
48C2×C3×C16—91no—
49C2×C3⋊C16—91no—
50#798—94no—
51C23×C32—31no—
52C22×C3×D3—22no—
53C2×C32×C4—22no—
54C2×C32×C4—31no—
55C2×C32×C4—32no—
56C2×C32×C4—32no—
57C2×C32×C4—42no—
58C2×C32×C4C4×D311yescharacteristic
59C2×C3×Dic3—22no—
60C3×C4×D3—24no—
61C3×C4×D3—24no—
62C3⋊(C3⋊C8)—32no—
63C3⋊(C3⋊C8)—64no—
64C3⋊(C3⋊C8)—124no—
65C3⋊(C3⋊C8)C4×D311yescharacteristic
66C3⋊(C3⋊C8)C22×D311yescharacteristic
67C3×C3⋊C8—62no—
68C3×C3⋊C8—62no—
69C2×M32—271noSylow
70C2×C33M3212yescomplemented
71C2×C33C22×C811yescharacteristic
72C32×D3—22no—
73C22×C3×C4—62no—
74C2×C4×D3#15711yescharacteristic
75C2×C3×C8—91no—
76C2×C3⋊C8—62no—
77C2×C3⋊C8—91no—
78C2×C3⋊C8—184no—
79C2×C3⋊C8—184no—
80C2×C3⋊C8—182no—
81C2×C3⋊C8—362no—
82D3×C8—92no—
83D3×C8—92no—
84C3×C16—92no—
85C3⋊C16—92no—
86C22×C32—22no—
87C22×C32—32no—
88C22×C32—42no—
89C22×C32—62no—
90C22×C32D3×C811yescharacteristic
91C2×C3×D3—28no—
92C2×C3×D3—24no—
93C32×C4—22no—
94C32×C4—22no—
95C32×C4—32no—
96C32×C4—42no—
97C32×C4—42no—
98C32×C4D3×C811yescharacteristic
99C32×C4C2×C4×D311yescharacteristic
100C3×Dic3—24no—
101C22×C8—271no—
102M32—274no—
103C2×C16—91no—
104C2×C16—271no—
105C33C2×M3211yescharacteristic, complemented, Sylow
106C23×C3—62no—
107C22×D3#62211yescharacteristic
108C2×C3×C4—22no—
109C2×C3×C4—42no—
110C2×C3×C4—64no—
111C2×C3×C4—64no—
112C2×C3×C4—62no—
113C2×C3×C4C2×#15711yescharacteristic
114C2×Dic3#62211yescharacteristic
115C4×D3#62212yes—
116C4×D3C2×#15712yes—
117C3×C8—91no—
118C3×C8—91no—
119C3⋊C8—62no—
120C3⋊C8—62no—
121C3⋊C8—92no—
122C3⋊C8—184no—
123C3⋊C8—364no—
124C2×C32—24no—
125C2×C32—22no—
126C2×C32—44no—
127C2×C32—42no—
128C2×C32—62no—
129C2×C32#79812yescomplemented
130C2×C32C2×D3×C811yescharacteristic
131C3×D3—28no—
132C22×C4—31no—
133C2×C8—91no—
134C2×C8—272no—
135C2×C8—272no—
136C2×C8—271no—
137C16—92no—
138C16—272no—
139C22×C3—22no—
140C22×C3—42no—
141C22×C3—64no—
142C22×C3—68no—
143C22×C3C2×#62211yescharacteristic
144C2×D3—22no—
145C2×D3C2×#62212yes—
146C3×C4—22no—
147C3×C4—22no—
148C3×C4—42no—
149C3×C4—42no—
150C3×C4—64no—
151C3×C4C2×#62211yescharacteristic
152C3×C4C22×#15711yescharacteristic
153Dic3C2×#62212yes—
154C32—22no—
155C32—42no—
156C32C2×#79811yescharacteristic, complemented
157C23—31no—
158C2×C4—31no—
159C2×C4—32no—
160C2×C4—32no—
161C2×C4D3×#15711yescharacteristic
162C8—91no—
163C8—91no—
164C8—272no—
165C2×C3—24no—
166C2×C3—22no—
167C2×C3—44no—
168C2×C3—42no—
169C2×C3—68no—
170C2×C3#6377312yescomplemented
171C2×C3C22×#62211yescharacteristic
172D3—22no—
173C22—32no—
174C22—62no—
175C22D3×#62211yescharacteristic
176C4—32no—
177C4D3×#62211yescharacteristic
178C4C2×D3×#15711yescharacteristic
179C3—22no—
180C3—42no—
181C3C2×#6377311yescharacteristic, complemented
182C2—62no—
183C2#1172272012yesdirect product factor
184C2C2×D3×#62211yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C22×C8
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C2×C8
prime radicalC22×C32×D3C8
cosocleC3×C3⋊(C3⋊C8)C23
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3⋊#3708C2semidirect
3. C2×C3×#3708C2semidirect
4. #11722720C2direct (N × H)
11. C3⋊#3708C22semidirect
12. C3×#3708C22semidirect
17. C2×#3708D3semidirect
39. #3708C2×D3semidirect
70. C2×C33M32semidirect
105. C33C2×M32semidirect
129. C2×C32#798semidirect
156. C32C2×#798semidirect
170. C2×C3#63773semidirect
181. C3C2×#63773semidirect
183. C2#11722720direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D3×C3⋊(C3⋊C8)2
C2×C3⋊#37082
C2×C3×#37082
#117227202
C2×#637733
C2×#7989